PF on Riemannian Manifolds

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Chapter 17

Particle Filtering on Riemannian Manifolds.

Application to Covariance Matrices Tracking

Hichem Snoussi

17.1 Introduction

Given a dynamical system characterized by a state-space model, the objective of

the online Bayesian filtering is the estimation of the posterior marginal probability

of the hidden state given all the observations collected until the current time. The

nonlinear and/or the non Gaussian aspect of the prior transition distributions and

the observation model leads to intractable integrals when evaluating the marginals.

Therefore, one has to resort to approximate Monte Carlo schemes. Particle filtering

[1] is such an approximate Monte Carlo method estimating, recursively in time, the

marginal posterior distribution of the continuous hidden state of the system. The

particle filter provides a point mass approximation of these distributions by drawing

particles according to a proposal distribution and then weighting the particles in order

to fit the target distribution.

The particle filter method is usually applied to track a hidden state belonging to an

Euclidean space. The most popular scheme is to sample the particles according to a

random walk around the previous particles. However, in some tracking applications,

the state may be constrained to belong to a Riemannian manifold. Recently, some

works have been dedicated to design algorithms adapted to the Riemannian mani-fold constraints, based on differential geometry tools: Gradient-descent algorithm on

Grassmann manifold for object recognition [2], statistical analysis of diffusion tensor

MRI[3], geodesic-based deconvolution algorithms [4], tracking principal subspaces

[5], algorithms in Stiefel and Grassman manifolds[6,7], statistical analysis on man-

ifolds [810], optimization on matrix manifolds[11, 12], and a general scheme for

tracking fast-varying states on Riemannian manifolds in [13]. This chapter is devoted

to the application of this differential-geometric framework to design efficient target

H. Snoussi(B)Charles Delaunay Institute,

UMR STMR 6279 CNRS, University of Technology of Troyes,

12, rue Marie Curie, 10010, Troyes, France

e-mail: [email protected]

F. Nielsen and R. Bhatia (eds.),Matrix Information Geometry, 427

DOI: 10.1007/978-3-642-30232-9_17, Springer-Verlag Berlin Heidelberg 2013


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428 H. Snoussi

tracking algorithms. We particularly consider the case where the observation noise

covariance is unknown and time-varying. The Bayesian filtering objective is thus to

jointly estimate the hidden target state and the time-varying noise covariance. As

the noise covariance is a positive definite matrix, the Euclidean space is not suitable

when tracking this covariance. Instead, one should exploit the differential geometricproperties of the space of positive definite matrices, by constraining the estimated

matrix to move along the geodesics of this Riemannian manifold. The proposed

sequential Bayesian updating consists thus in drawing state samples while moving

on the manifold geodesics.

The chapter is organized as follows: Sect. 17.2 is a brief introduction to the particle

filtering method on the Euclidean spaces. In Sect. 17.3,we describe some concepts

of differential geometry. In Sect. 17.4,we present a general scheme for the particle

filtering method on a Riemannian manifold. Section 17.5is dedicated to the design

of a particle filter jointly tracking a target state belonging to an Euclidean space anda time-varying noise covariance modeling the evolution over time of the sensing

system imperfections.

17.2 Bayesian Filtering on Euclidean Spaces

In this section, we briefly recall the particle filter method for filtering in nonlinear

dynamical systems characterized in Euclidean spaces. It is an approximate MonteCarlo method estimating, recursively in time, the marginal posterior distribution of

the continuous hidden state of the system, given the observations. The particle filter

provides a point mass approximation of these distributions. For more details and a

comprehensive review of the particle filter see[1,14,15].

The observed system evolves in time according to the following nonlinear dynam-

ics: xt px(xt | xt1, ut)

yt py (yt | xt, ut), (17.1)

where yt Rny denotes the observation at time t, xt Rnx denotes the unknown

continuous state, and ut Udenotes a known control signal. The probability dis-

tribution px(xt | xt1, ut)models the stochastic transition dynamics of the hidden

state. Given the continuous state, the observations yt follow a stochastic model

py (yt | xt, ut), where the stochastic aspect reflects the observation noise.

The Bayesian filtering is based on the estimation of the posterior marginal prob-

ability p(xt | y1:t). The nonlinear and the non Gaussian aspect of the transition

distributions leads to intractable integrals when evaluating the marginals. Therefore,

one has to resort to Monte Carlo approximation where the joint posterior distribution

p(x0:t | y1:t) is approximated by the point-mass distribution of a set of weightedsamples (called particles){x

(i )0:t, w

(i )t }

Ni =1:


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17 Particle Filtering on Riemannian Manifolds 429

pN(x0:t | y1:t)=

Ni =1

w(i )t x(i )0:t

(dx 0:t),

wherex(i )0:t(dx 0:t)denotes the Dirac function, and dx 0:tis the Lebesgue measure.Based on the same set of particles, the marginal posterior probability (of interest)

p(xt | y1:t)can also be approximated as follows:

pN(xt | y1:t)=

Ni =1

w(i )t x(i )t

(dxt),

Backward estimation of the marginal state probability is also possible given the

particles{x(i )

0:t+t , w

(i )

t+t }N

i =1

:

p(xt | y1:t+t )

Ni =1

w(i )t+tx(i )t

(dx t),

In the Bayesian importance sampling (IS) method, the particles {x(i )0:t}

Ni =1 are

sampled according to a proposal distribution(x0:t | y1:t)and {w(i )t }are the corre-

sponding normalized importance weights:

w(i )t

p(y1:t | x(i )0:t)p(x

(i )0:t)

(x(i )0:t | y1:t)

.

17.2.1 Sequential Monte Carlo

Sequential Monte Carlo (SMC) consists of propagating the trajectories {x(i )0:t}

Ni =1 in

time without modifying the past simulated particles. This is possible for the class ofproposal distributions having the following form:

(x0:t | y1:t)= (x0:t1 | y1:t1)(xt | x0:t1, y1:t).

The normalized importance weights are then recursively computed in time as:

w(i )t w

(i )t1

p(yt | x(i )t )p(x

(i )t | x

(i )0:t1)

(x(i )t | x

(i )0:t1, y1:t)

. (17.2)

For sake of clarity, one can adopt the transition prior as the proposal distribution:

(x(i )t | x

(i )0:t1, y1:t)= px(xt | xt1, ut)


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430 H. Snoussi

in which case the weights are updated according to the likelihood function:

w(i )t w

(i )t1p(yt | x

(i )t ).

The particle filter algorithm (depicted in pseudo-code Algorithm 17.1) consists of2 steps: the sequential importance sampling step and the selection step. The selection

(resampling) step replaces the weighted particles by unweighted particles in order to

avoid the collapse of the Monte Carlo approximation caused by the variance increase

of the weights. It consists of selecting the trajectories {x(i )0:t}with probabilitiesw

(i )t .

The trajectories with weak weights are eliminated and the trajectories with strong

weights are multiplied. After the selection step, all the weights are equal to 1 /N.

Algorithm 17.1Particle filter algorithm on an Euclidean space1: function PF(PP)

2: Initializationx(i )0 p0(x)

3: for t=1 toT do(Sequential importance sampling)

4: fori =1, ...,N do(sample from the transition prior)

5: x(i )t px(xt | x

(i )t1, ut)

6: set( x(i )0:t)= ( x

(i )t ,x

(i )0:t1)

7: end for

8: Update the importance weights

9: for i =1, ...,N do(evaluate and normalize the weights)

10: w

(i )

t p(yt | x(i)

t )11: end for

12: Resampling:

13: Select with replacement from { x(i)0:t}

Ni =1 with probability {w

(i)t } to obtain N particles

{x(i )0:t}

Ni =1

14: end for

15: end function

17.3 Differential Geometry Tools

In order to have a self-content framework, we devote this section to the introduction of

some differentiable geometry tools related to the concept of Riemannian manifolds.

These tools are necessary to the design of the particle filter on Riemannian manifolds.

For further details on Riemannian geometry, refer to[16].

First, we need to define a topological manifold as follows:

Definition 17.1 A manifoldM of dimension n, or n-manifold, is a topological space

with the following properties:(i) M is Haussdorff,

(ii) M is locally Euclidean of dimensionn , and

(iii) M has a countable basis of open sets.


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Fig. 17.1 Topological mani-

fold

U

2

M

1

Intuitively, a topological manifold is a set of points which can be considered locally

as a flat Euclidean space. In other words, each point p M has a neighborhood

Uhomeomorphic to an n-ball in Rn . Let be such an homeomorphism. The pair

(U,)is called a coordinate neighborhood: to p Uwe assign the n coordinates

1(p), 2(p), ..., n(p) of its image (p) in Rn (see Fig. 17.1). If p lies also in a

second neighborhood V, let(p)= [1(p), 2(p), ...,n (p)]be its correspondent

coordinate system. The transformation 1 on Rn given by:

1 : [1, ...,n] [1, ...,n ],

defines a local coordinate transformation on Rn from = [i ]to = [i ].In differential geometry, one is interested in intrinsic geometric properties which

are invariant with respect to the choice of the coordinate system. This can be

achieved by imposing smooth transformations between local coordinate systems

(see Fig. 17.2). The following definition of differentiable manifold formalizes this

concept in a global setting:

Definition 17.2 A differentiable (or smooth) manifoldM is a topological manifold

with a family U = {U,}of coordinate neighborhoods such that:

(1) theU cover M,(2) for any ,, if the neighborhoods intersection U U is non empty, then

1 and

1 are diffeomorphisms of the open sets(U U)and

(U U)ofRn ,

(3) any coordinate neighborhood (V,) meeting the property (2) with every

U, Uis itself in U

Tangent spaces. On a differentiable manifold, an important notion (in the sequel)

is the tangent space. The tangent space Tp(M) at a point p of the manifold M

is the vector space of the tangent vectors to the curves passing by the point p. It

is intuitively the vector space obtained by a local linearization around the point p.

More formally, let f : M R be a differentiable function on the manifold M

and : I M a curve on M, thedirectional derivativeof f along the curve

is written:


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432 H. Snoussi

1

1

U

2

2

M

U

1

Fig. 17.2 Differentiable manifold

Fig. 17.3 Tangent space on

the manifold

e1

M

2

1p

e2

d

dtf(t)=

fi

di

dt=

didt

i

f

where the derivative operator ei = i

can be considered as a vector belonging to

the tangent space at the point p. The tangent space is then the vector space spanned

by the differential operators i p:Tp(M)=

ci

i

p

| [c1, ..., cn ] Rn

,

where the differential operator ( i

)pcan be seen geometrically as the tangent vector

to the i th coordinate curve (fixing all j coordinates j =i and varying only the value

ofi ), see Fig. 17.3.

Vector fields and tensor fields. A vector field Xis an application M p Tp,

which assigns a tangent vector to each point of the manifold:

X : p S Xp Tp


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Fig. 17.4 Vector field on the

manifold

M

p

Xp = Xpii

The vector fieldXcan bedefined by its n component functions {Xi }n

i =1 (see Fig. 17.4).X isC (smooth) if and only if all its scalar components (Xi )are C

.

A tensor field A of type [q, r] is an application which maps a point p Mto

some multilinear mapping Ap fromTr

p toTq

p:

A : p ApAp : Tp Tp

r direct products

Tp Tp

q direct products

The types [0, r] and [1, r] are respectively called tensor fields of covariant degreerand tensor fields of contravariant degree 1 and covariant degree r. For example, a

scalar product is a tensor field of type[0, 2]:

Tp Tp R

(Xp, Yp)< Xp, Yp > .

Riemannian metric. For each point p in M, assume that an inner product p is defined on the tangent space Tp(M). Thus, a mapping from the points of

the differentiable manifold to their inner product (bilinear form) is defined. If thismapping is smooth, then the pair (M, < , >p)is calledRiemannian manifold(see

Fig. 17.5). The Riemannian metric is thus a tensor field g which is, according to a

coordinate system{}, defined by the positive definite matrices G p:

Gi j (p)=p

On a manifoldM, an infinite number of Riemannian metrics may be defined. The

metric does not thus represent an intrinsic geometric property of the manifold.

Consider now a curve: [a, b] (S, g), its length||||is defined as:


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434 H. Snoussi

M

p q

Fig. 17.5 Riemannian metric

|||| =

ba

||d

dt||dt=

ba

gi ji j dt

Affine connections. An affine connection is an infinitesimal linear relation p,p

between the tangent spaces of two neighboring points p and p (see Fig. 17.6). It

can be defined by its n 3 connection coefficients ki j (with respect to the coordinate

system[

i

]) as follows:

p,p ((j )p)= (j )p di (ki j )p(k)p

Let p andq be two points on M anda curve linking p and q. If the tangent

vectorsX(t)meet the following relation along the curve:

X(t+d t)= (t),(t+dt)(X(t)),

then, X isparallelalongand is aparallel translationon (see Fig. 17.7).

The covariant derivative of a vector field X along a curve is defined as the

infinitesimal variation betweenX(t) and the parallel translation ofX(t+h) T(t+h)to the space T(t) along . The parallel translation is in fact necessary in order to

consider the limit of the difference of two vectors belonging to the same vector

space. The vectors X(t) and X(t+d t) belong to different tangent spaces and the

quantity d X(t)= X(t+dt)X(t) may not be defined (see Fig. 17.8). The covariant

derivative D Xdt

forms then a vector field along the curveand can be expressed as a

function of the connection coefficients as follows:

D Xdt = ((t+dt),(t)(X(t+ dt)) X(t))/dt= {Xk(t) + i (t)Xj (t)(ki j )(t)}(k)(t)

(17.3)


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M

p

j

Tp

p

j

j

j

p,pdikijk

Tp

Fig. 17.6 Affine connections

(b)

(a)

(t)(t+dt)

X(a)

X(t) X(t+dt)

X(b)

(t) ,(t+dt)

Fig. 17.7 Parallel translation

The expression (17.3) of the covariant derivative along a curvecan be extended

to define the directional derivative along a tangent vectorDby considering the curve

whose tangent vector is D. The directional derivative, denoted by DX has the

following expression:

DX = Di {(iX

k)p + Xjp(

ki j )p}(k)p.

The covariant derivative along the curvecan then be written as:


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436 H. Snoussi

,p

X(t+dt)

(t+dt)

(t)

X(t)

X(t)

Tp

p

Fig. 17.8 Covariant derivative along a curve

D X(t)

dt= (t)X. (17.4)

Consider now two vector fields X and Y on the manifold M. The covariant

derivativeXY Tp(M) ofYwith respect to Xcan be defined by the following

expression:

XY = Xi {i Y

k +Y j ki j }k. (17.5)

The expression(17.5) of the covariant derivative can be used as a characterization

of the connection coefficients ki j . In fact, takingX =i and Y =j , the connection

coefficients are characterized as follows:

ij =ki jk.

A differentiable manifoldM is said to be flat if and only if there exists a coordinate

system[i ]such that the connection coefficients {ki j }are identically 0. This means

that all the coordinate vector fieldsi are parallel along any curveon M.

Riemannian connection. A Riemannian connection is an affine connection defined on a Riemannian manifold(M, g =)such thatX, Y,Z T(S), the

following property holds:


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Z < X, Y >=+ < X, ZY > (17.6)

where the left hand side of the equation means the differential operator Zapplied to

the scalar function< X, Y >on the manifold.

Letbe a curve on the manifold M and D Xdt et DYdt the covariant derivatives ofX and Y along respectively. According to the expression (17.4) of the covariant

derivative and the fact that the differential operator (t) consists of deriving with

respect tot, one has the following interesting identity concerning the variation of the

scalar product on the manifold with a Riemannian connection:

d

dt< X(t), Y(t) >=+ < X(t),

DY(t)

dt>

The above equation means that the scalar product is conserved under a paralleltranslation (D X(t)dt

= DY(t)dt

=0):

< (X), (Y) >=< X, Y > .

A particular example is the Euclidean space which is a flat manifold characterized

by a Riemannian connection.

Geodesics. A geodesic between two endpoints (a) and(b) on a Riemannian

manifold (M, g, ) is a curve : [a, b] M which is locally defined as the

shortest curve on the manifold connecting these endpoints. More formally, the defi-

nition of a geodesic is given by:

Definition 17.3 The parametrized curve(t)is said to be a geodesic if its velocity

(tangent vector) d/dtis constant (parallel) along, that is if it satisfies the condition

(D/dt)(d/dt)= 0, fora


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438 H. Snoussi

Ep(X)

M

Tp(M)

p

X

Fig. 17.9 Exponential mapping on the manifold

D(p, q)= ||g|| =

ba

gi ji j dt. (17.8)

The geodesic distance can also be defined as the shortest distance (over smooth

curves) between two points on the manifold endowed by a Riemannian connection.Exponential mapping. The exponential mapping is a central concept when

designing filtering methods on Riemannian manifolds. In fact, it represents an inter-

esting tool to build a bridge between an Euclidean space and the Riemannian man-

ifold. For a point p and a tangent vector X Tp(M), let : t = (t) be the

geodesic such that (0) = p and ddt

(0) = X. The exponential mapping of X is

defined as Ep(X) = (1). In other words, the exponential mapping assigns to the

tangent vector Xthe endpoint of the geodesic whose velocity at time t = 0 is the

vector X(see Fig. 17.9). It can be shown that there exist an neighborhood Uof 0 in

Tp(M)and a neighborhoodV ofpin M such that Ep |Uis a diffeomorphism fromU toV. Also, note that since the velocityd/dtis constant along the geodesic(t),

its length L from pto Ep(X)is:

L =

10

d

dtdt=

10

Xdt = X.

The exponential mappingEp(X) corresponds thus to the unique point on the geodesic

whose distance from pis the length of the vectorX.


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T(M) vt

xt=Ep(vt)

xt+1

xt1

M

vt+1

vt+2

Fig. 17.10 Markov chain on a Riemannian manifold

17.4 Particle Filtering on Riemannian Manifolds

17.4.1 General Scheme

The aim of this section is to propose a general scheme for the extension of the particle

filtering method on a Riemannian manifold. The hidden state x is constrained to lie

in a Riemannian manifold (M,g, ) endowed with a Riemannian metric g and

an affine connection . The system evolves according to the following nonlinear

dynamics: xt px(xt | xt1, ut), x M

yt py (yt | xt, ut), (17.9)

where the Markov chain (random walk) px(xt | xt1, ut) on the manifold M is

defined according to the following generating mechanism:

1. Draw a sample vton the tangent space Txt1M according to a pdf pv(.).

2. x is obtained by the exponential mapping of vt according to the affine

connection.

In other words, a random vector vt is drawn on the tangent space Txt1M

by the usual Euclidean random technics. Then, the exponential mapping allows

the transformation of this vector to a point xt on the Riemannian manifold. The

point xt is the endpoint of the geodesic starting from xt1 with a random initial

velocity vector vt. Figure 17.10illustrates the transition dynamics on a Riemannian

manifold M.As a generating stochastic mechanism is defined on the tangent space, the particle

filtering is naturally extended by means of the exponential mapping. It simply consists

in propagating the trajectories on the manifold by the random walk process, weighting


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440 H. Snoussi

the particles by the likelihood function and sampling with replacement. The proposed

general scheme is depicted in the pseudo-code Algorithm 17.2.

Algorithm 17.2Particle filter algorithm on a Riemannian manifold M1: function PF(PP)

2: Initializationx(i )0 p0(x)

3: for t=1 toT do(Sequential importance sampling)4: fori =1, ...,N do(sample from the random walk on M)

5: v(i )t pv (v) on Txt1M

6: x(i )t = Ex(i)t1

(v(i )t )

7: set( x(i )0:t)= ( x

(i )t ,x

(i )0:t1)

8: end for


10: for i =1, ...,N do(evaluate and normalize the weights)11: w

(i )t p(yt | x

(i)t )

12: end for

13: Resampling:

14: Select with replacement from { x(i)0:t}

Ni =1 with probability {w

(i)t } to obtain N particles

x(i )0:t}

Ni=1

15: end for

16: end function

17.4.2 Point Estimates

Based on particle trajectories{ x(i )0:T}, classical particle filtering algorithm provides a

simple way to approximate point estimates. In fact, any quantity of interesth(x)can

be estimated by its a posteriori expectation, minimizing the expected mean square

error. The empirical mean of the transformed particles h(x(i )t ) represents an unbiased

Mont-Carlo estimation of the a posteriori expectation. Averaging in the manifoldcontext is no more a valid operation: The empirical mean could be located outside

the manifold or the averaging itself does not have a meaning in the absence of a

summation operator on the manifold. In order to obtain a valid point estimate, one

should rather minimize the mean square error, where the error is evaluated by the

geodesic distance D on the manifold (17.8). Following the work of Frchet [17],

the point estimate can be defined by the intrinsic mean (also called Riemannian

barycenter). The intrinsic mean has the following expression:

xt =arg minxtME(D(xt,st))2 (17.10)=arg minxtM

(D(xt,st))

2p(st | y1..T)dst


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where the expectation operator is computed with respect to the a posteriori probability

density p(st | y1..T)and a Riemannian measuredinduced by the Riemannian met-

ric.

Remark 17.2 Note that the above estimator (17.10) does not yield the a poste-riori expectation in general. This is only the case when the manifold is flat with

respect to a Riemannian connection (Euclidean space). In fact, assume that[i ]

is a coordinate system corresponding to null Christoffel coefficients (ki j = 0).

A geodesic between 2 points p and q is then a straight line with coordinates

(p) + t((q) (p)). The geodesic distance D(p, q) (17.8) simplifies to a

quadratic distance

(i (p) i (q))2 leading to the usual Euclidean estimator

xt = E[xt | y1..T].

Computation of the point estimate (17.10) involves an integration operation (with

respect to st M and according to the posterior distribution) and a constrainedoptimization operation on the manifold M. The integral can be approximated (as in

the Euclidean case) by an empirical weighted sum of the geodesic distances applied

on the particles which are yielded within their weights by the particle filter Algorithm

17.2. The point estimate is then computed by the minimization of an approximated

expectation expression:

xt =arg min

xtM

N

i =1 w(i )t (D(xt,s

(i )t ))

2 (17.11)

where s(i )t andw

(i )t are the particles and their weights computed recursively by the

particle filter algorithm.

Concerning the constrained optimization in (17.11), more specific differential

geometric considerations should be taken into account. First, existence and unique-

ness of the intrinsic mean is shown by Karcher [18] when the manifold M has a

non positive sectional curvature. Second, a gradient descent like algorithm can be

designed on the manifold based on the exponential mapping which plays again a key

role in transferring Euclidean technics to a Riemannian manifold context. Denotingby J(xt)the objective function to be minimized with respect to xt,

J(xt)=

Ni =1

w(i )t (D(xt,s

(i )t ))

2,

a gradient flowx(l)t , starting from an initial guessx

(l)t and converging to the solution

xt, can be defined by moving in the direction of the opposite of the objective function

derivativeJ(x

(l)

t ). As the function derivative J(x

(l)

t ) lies in the tangent spaceTx

(l)t

(M), the exponential mapping can be used to map the opposite derivative vector

tothenextpointx(l+1)t . The gradient-like descent algorithm is then derived as follows:


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442 H. Snoussi

Tx(l)t

(M)

M

x(l+1)t =E

x(l)t

(J)J x(l)t

Fig. 17.11 A gradient descent step on a Riemannian manifold

x(l+1)t =Ex(l)t

(J(x(l)t )) (17.12)

Figure 17.11illustrates an iteration of the gradient descent algorithm on a Rie-

mannian manifold.

Remark 17.3 The implementation of the proposed particle filter algorithm on the

manifoldM and the computation of the point estimates require an explicit expressionof the exponential mapping. In other words, solving geodesic differential Eq. (17.7)

conditions the feasibility of the manifold version of the particle filter algorithm.

Explicit analytic expressions are only available in some cases. Among these cases,

one can find:

Flat manifolds (with constant null connections ) where geodesics are straight

lines.

The set of Gaussian probability densities with fixed mean: This example will play

a central role in the next Sect. 17.5devoted to the target tracking application.

17.5 Application to Tracking with Unknown Time-Varying State

Covariance

This section is devoted to the application of the above developed framework to design

an efficient target tracking algorithm. The target state is assumed to be observed

according to the general system (17.9), where the observation equation is assumed

to have a general form depending on the sensing model. Concerning the transition

dynamics px(xt | xt1), we adopt a mean-scale mixture model. According to this

model, the hidden statext Rnx (belonging to an Euclidean space) has a Gaussian

distribution with a random mean tand a random covariance matrix t. The mean


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follows a Gaussian random walk reflecting the time correlation of the system tra-

jectory and the covariance matrix follows a Generalized Gaussian random walk

on the Riemannian manifold of positive definite matrices S+. The transition prior is

thus defined by an augmented Markov state(t,t,x t)as follows:t N(t |t1, )

t GN(t | t1,)

xt N(xt |t,t)

(17.13)

Contrary to the usual assumption of a constant known covariance, the case of

a stochastic varying state covariance represents an elegant way to deal with abrupt

changes in state trajectories. In fact, this parametric family is able to describe the fat

tails and the skewness of the regression modelpx(xt | xt1, ut). In fact, the resulting

regression model can be written in an integral form:

px(xt | xt1, ut)=

p(t,t | xt1)N(xt; t,t)dtdt

showing that the regression model is an infinite continuous mixture of Gaussian mod-

els. This model can be considered as the extension of the scalar Generalized Hyper-

bolic distributions introduced by Barndorff-Nilesen [19] to a multivariate regression

context.

Before defining the Generalized Gaussian random walkGNof covariance matri-

ces and the particle filter algorithm, we introduce hereafter the expressions of the

Riemannian metric and the Riemannian connections in the special case of the Rie-

mannian manifold S+. Closed forms for the geodesic curves and geodesic distances

are also obtained, providing an efficient implementation of the general particle filter

scheme and the point estimate computation proposed in the previous Sect. 17.4.

17.5.1 Space of Positive Definite Matrices

A positive definite matrix can represent the covariance matrix of a zero-mean mul-

tivariate normal distributionN(0,). Therefore, the set of positive definite matrices

S+can be identified with the statistical model of multivariate Gaussian distributions.

Analyzing the geometric structure of parametric statistical models based on differ-

ential geometric tools was first suggested by Rao in 1945[20]. Since then, many

works have been dedicated to the analysis of statistical manifolds in the light of dif-

ferential geometry. In particular, the seminal work of Amari[21] led to a significant

advancement in this new field.

In the work of Rao [20], it was suggested that the Fisher Information matrixprovides a natural Riemannian metric on statistical manifolds. Denoting by p(.| )a point of a statistical parametric manifold, the Fisher metric is expressed as:


18/23

444 H. Snoussi

gi j ()= E

log p(x |)

i

log p(x |)

j

=

X

log p(x |)

i

log p(x |)

jp(x |)d(x)

Assuming that the connection is symmetric and torsion free, the Riemannian con-

nection can be uniquely defined given the Riemannian metric. It is determined by

the following expression of its Christoffel symbols of the second kind:

ki j = gkm i j m =

1

2gkm (

gj m

i+gi m

jgi j

k),

whereg km are the elements ofg 1 (inverse Fisher matrix).

Applying the above expressions to the case of the manifoldS+ = {N(0;), >

0}of multivariate normal distributions, a closed form for the Riemannian metric is

derived as follows:

g(Bi,j ,Bk,l )=< Bi,j ,Bk,l >= 1

2trace{1Bi,j

1Bk,l }, i, j, k, l =1..nx

(17.14)

where{Bi,j }is the canonical basis of the tangent space TS+which coincides with

the space of real symmetric matrices S. The matrix Bi,j is expressed as follows:

Bi,j = 1ii , i = j

1i j

+1j i

, i = j

with 1i j standing for the matrix with 1 at rowi and column j and 0 elsewhere.

A closed form for the geodesic curve is derived in [22] for the general case of

multivariate normal distributions. Here, we only consider the case of multivariate

normal distributions with a fixed zero-mean. Denoting by (r s ), r s, s =1..nx the

components of the positive definite matrix and using the explicit expressions of the

Riemannian metric and connections, the Euler-Lagrange equations (17.7) reduces to

the following matrix-form differential equation:

d2(t)

dt2

d(t)

dt

1(t)d(t)

dt=0,

leading to the following closed form of a geodesic (t)starting from (0)with an

initial velocity (0):

(t)= (0)1/2 exp [tx](0)1/2, (17.15)

where the matrix x T(0)S+ is defined as,

(0)= (0)1/2x(0)1/2,

and exp [.] stands for the usual matrix exponentiation.


19/23


The geodesic distance between two matrices1and 2 D (length of the geodesic

curve between 1 and 2) has an explicit analytic form as well:

D(1,2)= Ln(1/21 2

1/21 )

=1

2

nxi =1log

2(i )

where Ln(.)is the matrix logarithm defined by Ln(exp[A])= Afor ASandiare the eigenvalues of the matrix

1/21 2

1/21 .

17.5.2 Particle Filter Implementation

Given the explicit analytic expression of the geodesic curve(17.15)for the space S+of positive definite matrices, the generating mechanism of a Gaussian random walk

GN(t | t1,)(as defined in Sect. 17.4), has the following steps:

1. Sample a Gaussian symmetric velocity matrix B S with a precision

( nx(nx+1)2

nx(nx+1)2

matrix): B N(0;)

2. The next matrix tis then obtained by:

t =Et1 (B)= 1/2t1exp

1/2t1 B

1/2t1

1/2t1

The proposed particle filter jointly estimate the hidden target positionx t(belong-

ing to an Euclidean space) and its covariance t (belonging to the Riemannian

manifold S+) as follows (see Algorithm 17.3):

1. Propagate the trajectories ((i )0:t1,

(i )0:t1,x

(i )0:t1) by generating the samples

((i )t ,

(i )t ,x

(i )t )according to the prior model (17.13).

2. Update the importance weights which are proportional to the likelihood function.

The explicit solution of the geodesic distance allows also the implementation of

the intrinsic mean for the tracking of the covariance matrix. In fact, approximatingthe expected error by the empirical weighted sum of geodesic distances, the point

estimate is defined as follows:

t = arg minS+

Ni =1

w(i )t (D(,

(i )t ))

2

= arg minS+

Ni =1

w(i )t

1

2trL n2(1/2

(i )t

1/2)

The gradient of the objective function, belonging to the tangent space TS+, has

the following expression:


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446 H. Snoussi

J()=

N

Ni =1

Ln(((i )t )

1) (17.16)

Given the explicit expression of both the gradient (17.16) and the exponentialmapping (17.15) on the manifoldS+, the gradient-descent algorithm scheme defined

in Sect. 17.4for the computation oft is efficiently implemented. Given an initial

guess (0)

, a gradient flow (l)

evolving towards the solution is defined as follows:

(l+1)

=E

(l) (J((l)

))

= ((l)

)1/2 exp

((l)

)1/2J((l)

)1/2

((l)

)1/2

Algorithm 17.3Particle filter algorithm for tracking

1: function PF(PP)

2: Initialization((i )0 ,

(i)0 ,x

(i )0 ) p0

3: for t=1 toT do(Sequential importance sampling)4: fori =1, ...,N do(sample from the random walk prior)

5: (i )t px(t |

(i )t1)

6: B(i )t N(0;) on T(i )t1

S+

7: (i )t = E(i )t1

(B(i )t )

8: x(i )t N((i)t ; (i)t )

9: set (i)t = (

(i )t ,

(i )t ,x

(i )t ) (Augmented state)

10: set ((i )0:t) = (

(i )t ,

(i )0:t1)

11: end for


13: for i =1, ...,N do(evaluate and normalize the weights)

14: w(i )t p(yt | x

(i)t )

15: end for

16: Resampling:

17: Select with replacement from{(i )0:t}

Ni =1 with probability{w

(i )t }

18: end for

19: end function

17.5.3 Simulation Results

We consider the tracking of a target moving over a 2-D field. The statex t = [xpt ,x

vt]

is formed by the position and the velocity of the target. For simplicity, we assume a

kinematic parametric model for the transition dynamics of the hidden state:


21/23


40 45 50 55 60 650

10

20

30

40

50

60

70

80

90True Trajectoy

Estimated trajectory

Rangebearing sensor

Fig. 17.12 Target tracking with unknown noise covariance. Note that only the selected nodes are

plotted in the figure

x

pt

xvt

=

1 0 Ts 0

0 1 0 Ts0 0 1 0

0 0 0 1

xpt1

xvt1

+

T2s /2 0

0 T2s /2

Ts 0

0 Ts

ut

where the sampling interval is Ts = 0.1s and ut is a zero-mean white Gaussian

noise.

The observations are obtained through a network of 400 range-bearing sensors

deployed randomly in the field under surveillance. At each time t, a selected node

(according to the proximity to the target) obtains an observation of the target position

through a range-bearing model:

yrtyt= psm xt+0.5arctan s2x2

s1x1

+ tvtwheresm = (s1, s2)and xt = (x1,x2)are the node and the target positions at time

t, p(set to 10) is the energy emitted by the target (measured at a reference distance

of 1 meter) and vtis a white Gaussian random vector. The corrupting noise has a

covariance tevolving in time as depicted in Fig. 17.13: constant for the first T/4s,

increasing with a linear slope for T/2s and constant for the lastT/4s.

The particle filter is applied to jointly estimate the target position and the noise

covariance matrix. Figure 17.12 illustrates the target tracking performances. Thetrajectory of the target is recovered with a mean square error of 0.39 m. Figure17.13

illustrates the performance of the algorithm to online track the covariance variation

over time. Note that, despite their fluctuation, the estimated covariance elements


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448 H. Snoussi

50 100 150 200 250 300 350 400 450 5000.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4 True valueEstimated value

11

50 100 150 200 250 300 350 400 450 5000.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4 True valueEstimated value

12

50 100 150 200 250 300 350 400 450 5000.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (s)

True valueEstimated value

21

50 100 150 200 250 300 350 400 450 5000.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (s)

Estimated valueTrue value

22

Fig. 17.13 Online estimation of the noise covariance elements

follow the tendency of the true covariance elements. The fluctuation of the estimated

noise covariance is mainly due to the fact that the data are less informative with

respect to the covariance matrix. In fact, unlike the target position estimation, the

online estimation of the covariance t is an ill-posed problem based on only one

observation yt. The success of the algorithm to approximately recover the tendency

of the covariance matrix is due to the Markov prior regularization defined by the

Generalized Gaussian random walkGN

(

t |

t1,

) defined in the previoussubsection.

17.6 Conclusion

A differential-geometric framework is proposed to implement the particle filtering

algorithm on Riemannian manifold. The exponential mapping plays a central role

in connecting the manifold-valued particles to the samples generated on the tangent

space by the usual random generating techniques on Euclidean spaces. The proposed

algorithm has been applied to jointly track the target position with the time-varying

noise covariance.


23/23


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PF on Riemannian Manifolds

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